In the old counterfactual mugging problem, agents who precommit are trading utilities across possible worlds, each world having a utility-gain called a prior that expresses how much the agent wants its utilities to lie in those worlds instead of silly ones. From that perspective, it's true that nothing in reality will be different as a result of the agent's decision, just because of determinism, but the agent is still deciding what reality (across all possible worlds) will look like, just like in Newcomb's problem.
So when I read in Nesov's post that "Direct prediction of your actions can't include the part where you observe that the digit is even, because the digit is odd", what I'm really seeing is someone saying, "I give zero weight to possible worlds in which math doesn't work sensibly, and tiny weights to worlds in which math does work, but my confusion or the conspiring of a malicious / improbable / senseless / invaluable universe cause me to think it does not."
One of the reasons why I think possible worlds of the first kind (different causal / programmatic histories but the same underlying ontology-stuff) are valuable / real, is that we sort of know how to calculate their properties using causal networks or timeless networks or whatever kind of networks you get when you combine the not-quite specified mathematical machinery in TDT with UDT. Our ability to calculate their properties reifies them, opens them up to interacting with this world even more directly via simulation.
The next step seems to be to ask, "for agents that do care about those impossible possible worlds, how would they act?" If omega is choosing in a way that can be computed in our world, using our math (and somehow that other universe and our calculations don't explode when it gets to the contradiction (or it does! I suppose you can care about worlds where math explodes, even if I can't visualize them)), then we can simulate his reasoning in all respects save the identify of the logical fact in question, and use that to calculate which behaviour maximizes the utility across possible worlds via their dependence on our decision.
So in the example problem, if a valuer of contradictory worlds has roughly equal priors for both the world we're examining and the other world in which she find herself where the digit was even (the impossible one, which isn't impossible for her, because it wasn't assigned zero prior weight), then sure, she can go ahead and give up control. That's of course assuming that she has an expectation the staple maximizer will reciprocate in the impossible world, which you didn't spell out in your post, but that dependence on decisions is standard for counterfactual mugging problems. Please correct me if that's not the intended setup.
As as aside, this comment feels silly and wrong; an example of diseased thoughts unconnected with reality. It reminds me a bit of Greg Egan's short story Dark Integers. I would really love to see a more sensible interpretation that this.
While I haven't given it much though outside the context of fiction, one could adopt the point of view/vocabulary of this being "the level 5 tegmark mutiverse".
Now, if that is true in any sense, it's probably a much less literal one, and not based on the same reasoning as the other four, but it might still be an useful heuristic for humans.
Another interesting note: By default my brain seems to assume utility is linear with paperclips when considering say different Everett branches, but the logarithm of it when considering logical uncertainty. That's kinda odd and unjustified, but the intuition might have some point about humans utility function.
Suppose you wake up as a paperclip maximizer. Omega says "I calculated the millionth digit of pi, and it's odd. If it had been even, I would have made the universe capable of producing either 1020 paperclips or 1010 staples, and given control of it to a staples maximizer. But since it was odd, I made the universe capable of producing 1010 paperclips or 1020 staples, and gave you control." You double check Omega's pi computation and your internal calculator gives the same answer.
Then a staples maximizer comes to you and says, "You should give me control of the universe, because before you knew the millionth digit of pi, you would have wanted to pre-commit to a deal where each of us would give the other control of the universe, since that gives you 1/2 probability of 1020 paperclips instead of 1/2 probability of 1010 paperclips."
Is the staples maximizer right? If so, the general principle seems to be that we should act as if we had precommited to a deal we would have made in ignorance of logical facts we actually possess. But how far are we supposed to push this? What deal would you have made if you didn't know that the first digit of pi was odd, or if you didn't know that 1+1=2?
On the other hand, suppose the staples maximizer is wrong. Does that mean you also shouldn't agree to exchange control of the universe before you knew the millionth digit of pi?
To make this more relevant to real life, consider two humans negotiating over the goal system of an AI they're jointly building. They have a lot of ignorance about the relevant logical facts, like how smart/powerful the AI will turn out to be and how efficient it will be in implementing each of their goals. They could negotiate a solution now in the form of a weighted average of their utility functions, but the weights they choose now will likely turn out to be "wrong" in full view of the relevant logical facts (e.g., the actual shape of the utility-possibility frontier). Or they could program their utility functions into the AI separately, and let the AI determine the weights later using some formal bargaining solution when it has more knowledge about the relevant logical facts. Which is the right thing to do? Or should they follow the staples maximizer's reasoning and bargain under the pretense that they know even less than they actually do?
Other Related Posts: Counterfactual Mugging and Logical Uncertainty, If you don't know the name of the game, just tell me what I mean to you